Transcript Assignment Model Example 6

Department of Business Administration
SPRING 2007-08
Management Science
by
Asst. Prof. Sami Fethi
© 2007 Pearson Education
1
Ch 6: Transportation and Assignment
Chapter Topics
The Transportation Model
Solution of a Transportation Problem
The Assignment Model
Solution of an Assignment Problem
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved.
2
Ch 6: Transportation and Assignment
Overview
Part of a larger class of linear programming problems
known as network flow models.
Possess special mathematical features that enabled
development of very efficient, unique solution methods.
Methods are variations of traditional simplex procedure.
Detailed description of methods is contained in CD-ROM
Module B, Transportation and Assignment Solution
Methods.
Text focuses on model formulation and solution with Excel
and QM for windows.
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved.
3
Ch 6: Transportation and Assignment
The Transportation Model
Characteristics
A product is transported from a number of sources to a
number of destinations at the minimum possible cost.
Each source is able to supply a fixed number of units of the
product, and each destination has a fixed demand for the
product.
The linear programming model has constraints for supply at
each source and demand at each destination.
All constraints are equalities in a balanced transportation
model where supply equals demand.
Constraints contain inequalities in unbalanced models
where supply does not equal demand.
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved.
4
The Transportation Model
Characteristics
Ch 6: Transportation and Assignment
Transportation modelling is an iterative procedure for solving
problems that involve minimizing the cost of shipping products
from a series of sources to a series of destinations.
 Origin Points (or sources) can be factories, warehouses, car
rental agencies like AVIS or any other points from which goods
are shipped.
 Destinations are any points that receive goods.
Transportation models are useful when considering alternative
facility locations. The choice of a new location depends on
which will yield the minimum cost for the entire system
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved.
5
Ch 6: Transportation and Assignment
The Transportation
Model Characteristics
To use the transportation model we need to know the following:
 The origin points and the capacity or supply per period at each
 The destination points and the demand per period at each
 The cost of shipping one unit from each origin to each
destination
 The transportation problem can be regarded as a generalization of the
assignment problem. In an assignment problem we have a number of origins
each processing one item, and the same number of destinations each
requiring one item and are asked to empty the origins and fill the
destinations in such a way the Total Effectiveness is optimized. In a
transportation problem we have “m” origins, with origin “i” processing “ai”
items and “n” destinations (possibly different number from “m”) with
destination “j” requiring “bj” items, and with
ai = bj.
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved.
6
Ch 6: Transportation and Assignment
The Transportation Model Characteristics
We are given the m.n costs associated with shipping one
item from any origin to any destination, and are asked to
empty the origins and fill the destinations in such a way that
the total cost is minimized.
The assignment problem is seen to be a special case of
transportation problem in which m = n, all the ai and bj are
unity.
The iterative routine to be presented as the mode of the
solution of the transportation problem involves a succession
of “feasible solutions” of gradually diminishing cost;
eventually the minimum achievable cost is reached, and the
corresponding feasible solution is optimal.
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved.
7
Transportation Model Example1
(1)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved.
8
Transportation Model Example
(2 )
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved.
9
Transportation Model Example
(3)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 10
Transportation Model Example
(4)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 11
Ch 6: Transportation and Assignment
Transportation Model Example
(5)
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved. 12
Transportation Model Example
(6)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 13
Transportation Model Example 2
(1)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 14
Transportation Model Example
(2)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 15
Transportation Model Example
(3)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 16
Transportation Model Example
(4)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 17
Ch 6: Transportation and Assignment
The Assignment Model
Characteristics
Special form of linear programming model similar to the
transportation model.
Supply at each source and demand at each destination
limited to one unit.
In a balanced model supply equals demand.
In an unbalanced model supply does not equal demand.
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved. 18
The Assignment Model
Characteristics
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 19
Assignment Model Example 1
(1)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 20
Assignment Model Example
(2)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 21
Assignment Model Example 2
(1)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 22
Assignment Model Example 3
(1)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 23
Assignment Model Example 4
(1)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 24
Assignment Model Example 4
(2)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 25
Assignment Model Example
(3)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 26
Assignment Model Example
(4)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 27
Ch 6: Transportation and Assignment
Assignment Model Example 5
(1)
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved. 28
Assignment Model Example 5
(2)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 29
Ch 6: Transportation and Assignment
Assignment Model Example 6
(1) (minimize)
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved. 30
Assignment Model Example 6
(2) (minimize)
Operations Research
Ch 6: Transportation and Assignment
© 2007/08, Sami Fethi, EMU, All Right Reserved. 31
Ch 6: Transportation and Assignment
Assignment Model Example 6
(3) (minimize)
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved. 32
Ch 6: Transportation and Assignment
Assignment Model Example 6
(4) (minimize)
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved. 33
Ch 6: Transportation and Assignment
End of chapter
Operations Research
© 2007/08, Sami Fethi, EMU, All Right Reserved. 34